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34754 results in Cambridge Textbooks

Page 1644 of 1738

Mathematical Methods and Models for Economists
  • Angel de la Fuente
  • Published online:
    05 June 2012
    Print publication:
    28 January 2000
  • This book is intended as a textbook for a first-year PhD course in mathematics for economists and as a reference for graduate students in economics. It provides a self-contained, rigorous treatment of most of the concepts and techniques required to follow the standard first-year theory sequence in micro and macroeconomics. The topics covered include an introduction to analysis in metric spaces, differential calculus, comparative statics, convexity, static optimization, dynamical systems and dynamic optimization. The book includes a large number of applications to standard economic models and over two hundred fully worked-out problems.

  • Preface

  • Angel de la Fuente
  • Book:
    Mathematical Methods and Models for Economists
    Published online:
    05 June 2012
    Print publication:
    28 January 2000, pp xi-xii

  • Summary

    Much of the time of the average graduate student in economics is spent learning a new language, that of mathematics. Although the investment does eventually pay off in many ways, the learning process can be quite painful. I know because I have been there. I remember the long nights spent puzzling over the mysteries of the Hamiltonian, the frustration of not understanding a single one of the papers in my second macroeconomics reading list, the culture shock that came with the transition from the undergraduate textbooks, with their familiar diagrams and intuitive explanations, into Debreu's Theory of Value, and my despair before the terse and incredibly dry prose of the mathematics texts where I sought enlightenment about the arcane properties of contractions.

    This book is an attempt to make the transition into graduate economics somewhat less painful. Although some of my readers may never believe me, I have tried to do a number of things that should make their lives a bit easier. The first has been to collect in one place, with a homogeneous notation, most of the mathematical concepts, results, and techniques that are required to follow the standard first- and second-year theory courses. I have also tried to organize this material into a logical sequence and have illustrated its applications to some of the standard models. And last but not least, I have attempted to provide rigorous proofs for most of the results as a way to get the reader used to formal reasoning.

  • 8 - Some Applications to Microeconomics

  • from PART II - STATICS
  • Angel de la Fuente
  • Book:
    Mathematical Methods and Models for Economists
    Published online:
    05 June 2012
    Print publication:
    28 January 2000, pp 325-388

  • Summary

    The first part of this book has covered most of the mathematical tools required for analysis of static economic models. In this chapter we will discuss some applications of this material to a number of microeconomic models. Our goal will not be to provide a comprehensive treatment of a set of topics generally covered in the standard first-year graduate sequence in microeconomics, but only to illustrate the usefulness of the techniques we have developed and to introduce the reader to the general logic of modelbuilding in economic theory.

    We began Chapter 7 with the observation that the “postulate of rationality” – the assumption that individuals have well-defined and consistent preferences and act accordingly – is central in (neoclassical) economics as a source of regularity in individual behavior that makes prediction possible, at least in principle. We then claimed that this postulate led naturally to the modeling of individual decision-making as the outcome of a constrained optimization problem, and we devoted a fair amount of time to studying the “technology” required for solving such problems. Section 1 of this chapter backtracks a little. We consider a standard consumer and discuss how his preferences can be represented by a binary relation and how this relation can be used to construct a utility function. Section 2 then analyzes the behavior of this consumer when he faces market-determined prices for the commodities he wants to purchase with his (exogenously given) income.

  • Appendix on Notation

  • Harold Jeffreys, University of Cambridge and St John's College, Cambridge, Bertha Jeffreys, Girton College, Cambridge
  • Book:
    Methods of Mathematical Physics
    Published online:
    12 September 2018
    Print publication:
    18 November 1999, pp 706-710

  • Summary

    The difficulty of learning mathematical physics is much increased by confusion of notation, especially the overworking of certain letters and the introduction of awkward sign conventions. The only criterion usually recognized is conformity with ‘standard practice’. Unfortunately standard practice is not unique and students are put to much unnecessary trouble by having to accustom themselves to work with different conventions in rapid succession. Research workers in border-line subjects are also inconvenienced by finding the usual symbols in one subject pre-empted for different meanings in another.

    The following principles are important in choosing conventions:

  • (1) Complications should be reduced to a minimum. Negative signs should not be introduced without good reason.

  • (2) Genuine physical differences should be recognized as such and not disguised as conventions; attempts to disguise them always lead to later difficulties that should have been forestalled.

  • (3) Where a mathematical theory has applications in several subjects the notation should be such that it can be carried over into those subjects unchanged; so far as possible it should not use symbols already used with other meanings in those subjects.

    • The outstanding difficulty of notation at present is the ambiguous use of Y and φ. V is used for potential energy, which is a property of a complete system, but also for the various potential functions, which are functions of position within the system. It is also used for Hamilton's characteristic function, and, in hydrodynamics, for a component of the velocity at a great distance.

Page 1644 of 1738